This is far from an actual answer, but it suggests support for the idea.
If a construction were to proceed here, a few first steps would be:
- to find a meaningful example of a paraconsistent logic that allows us to think clearly about it and judge the success of our project as it proceeds
- to answer Niel de Beaudrap's concern as to whether there is a compelling topology on the space of its decisions
- to think about what kinds of mappings from that topology onto set-theory or measures seems to capture our 'degree of consistency'.
It seems to me that the primary model of paraconsistency that we use on a daily basis is morality. People who reason about morality in one very conventional way admit contradictions in their moral axioms all the time. Then they work outward on difficult issues from a highly conflicted position to one which achieves an acceptable level of consistency by taking in more and more perspective.
So a good model, in my head, would be a modal logic on the mode of "ought", with a common set of conflicting axioms, and degrees of consistency.
If you don't agree that morality works that way, Common Law explicitly tries to. Laws depend upon past application, and precedent matters more the closer it is or the more settled it is and the whole point of each judgement is to maintain or increase the internal consistency of the system, hoping that consistent and not offensive eventually devolves on just.
So I think there probably is a clearly articulated topology here, with open sets being something like 'must be considered to decide'. If you start from a very clear prescriptive cult doctrine and some Bayesian version of the 'ought' logic, or a rigidly statutory version of Common Law with an 'expert system' notion of an AI judge, you could probably present an approximation to that topology as simply as a graph.
It would not be a simple graph, and it would have to allow for degrees, regressions, etc., it may need probabilistic or bi-measured edges, etc. (to capture recency and settledness?) But, in concept, each statute or tenet applies to bodies of other decisions in a vaguely hierarchical way.
A second potential example might be causal flow or 'collapse' in a quantum environment limited by relativity or some other localizing principle. [Out comes the harp /] If you adopt my favorite notion of time as accumulated entropy, I can imagine making this a nice clean topology like a metric space on a manifold, by considering the merger of the local balances on entropy as the wavefronts sweep outward from separate decisions to find a global consensus of the direction in which entropy increases. The measure on consistency could be driven by "How long do I have before I am globally observable?"